Integral in Normal Distribution

Normal distribution is defined by the so-called "hat function": \(\displaystyle f (x)=\frac{1}{2\sqrt{\pi}}e^{-\frac{1}{2}x^{2}}\) The coefficient \(\displaystyle \frac{1}{2\sqrt{\pi}}\) is so chosen as to insure that \(\displaystyle \int_{-\infty}^{\infty} f(x)dx=1.\)

To see how this come about, I'll compute the integral \(\displaystyle I=\int_{-\infty}^{\infty} e^{-x^{2}}dx\). The idea is to convert the integral to a double integral by squaring and subsequently change to polar coordinates: